Simulating the bulk storage of foodstu€s

Simulating the bulk storage of foodstu€s

Yongfu Xu, Dean Burfoot

Silsoe Research Institute, Wrest Park, Silsoe, Bedfordshire MK45 4HS, UK Received 19 July 1998; accepted 10 October 1998

Abstract

This paper presents a transient three-dimensional CFD model of heat and mass transfer in porous bulks of particulate foodstu€s.

The mass, momentum and energy equations were solved to predict the air ¯ows, temperature and moisture changes of the air and solids. The interaction between the air ¯ows and the porous media were described by the Ergun equation. Moisture di€usion and heat transfer within the solids was predicted assuming the solids to be spheres, and the heat of respiration was included in the model as an empirically derived function of temperature. The predicted and measured temperature pro®les across a bed of potatoes with a height of 2.4 m and a diameter of 0.7 m during forced cooling, from 15.5°C to 6°C, show similar spatial and temporal variations with a maximum di€erence of 1.4°C. Predicted weight loss of the bed in a 4-day test period was within 5% of the measured value. Ó 1999 Elsevier Science Ltd. All rights reserved.

Keywords: CFD simulation; Heat transfer; Mass transfer; Evaporation; Cooling; Food storage

Notation

A surface area of a potato, m²

a speci®c surface area of a potato medium, m²m^ÿ3 c speci®c heat, J kg^ÿ1K^ÿ1

D di€usivity of vapour in air, m²s^ÿ1

Deff e€ective di€usivity of water through particle skin, m²s^ÿ1

Dp di€usivity of water through solid particles, m²s^ÿ1 dp diameter of solid particles, m

H enthalpy per unit mass, J kg^ÿ1 h latent heat, J kg^ÿ1

hm mass transfer coecient, m s^ÿ1 ht heat transfer coecient, W m^ÿ2K^ÿ1 K permeability of a porous medium, m² k thermal conductivity, W m^ÿ1K^ÿ1 L^ characteristic length of a porous medium,

L^ˆ dpc /(1)c )

m mass transfer rate, kg m^ÿ3s^ÿ1 Fht heat ¯ux across a particle surface, J s^ÿ1 Fms mass ¯ux across a particle surface, kg s^ÿ1 Nu Nusselt number, Nu ˆ htdp/k

Pr Prandtl number, Pr ˆ m/a

P pressure, Pa

q heating by respiration and conduction through the bulk, J kg^ÿ1s^ÿ1

q1 heat of respiration, J kg^ÿ1s^ÿ1 Ri resistance components, N m^ÿ3

Ra Rayleigh number, Ra ˆ gKdpb(Tps)Ta)/(am) Ram mass transfer Rayleigh number,

Ramˆ gKdpbc(qv;ps)qv;a)/(Dm) Re Reynolds number, Re ˆ Udp/m

r radius, m

Sc Schmidt number, Sc ˆ m/D Sh Sherwood number, Sh ˆ hmdp/D

t time, s

T temperature, K

ui ¯uid velocity components, m s^ÿ1 U ¯uid velocity, m s^ÿ1

Xi cartesian coordinates

Y mass fraction of vapour, kg water/kg air Greek

a thermal di€usivity, m²s^ÿ1

b volumetric thermal expansion coecient, K^ÿ1 bc composition expansion coecient of air/vapour,

m³kg^ÿ1

c porosity of the porous medium l ¯uid viscosity, kg m^ÿ1s^ÿ1 m kinematic viscosity, m²s^ÿ1

q density, kg m^ÿ3

qv vapour mass concentration, kg m^ÿ3 v moisture content, kg water/kg solid material v0 initial moisture content, kg water/kg solid mate- s rialshare stress, N m^ÿ2

w water activity (partial pressure of the vapour/

saturated vapour pressure) Subscripts

a air/vapour ¯ow

bp bulk solid particles, e.g. potatoes

map mass-averaged property of the solid particles inside a cell

*Corresponding author.

0260-8774/99/$ ± see front matter Ó 1999 Elsevier Science Ltd. All rights reserved.

PII: S 0 2 6 0 - 8 7 7 4 ( 9 8 ) 0 0 1 3 9 - 3

1. Introduction

Most foodstu€s, such as grain, potatoes and onions, are stored in piles, silos or boxes. Key factors in the economic success of storage are minimising weight loss by evaporation, and restricting energy usage while avoiding disease and maintaining product quality. All of these factors are in¯uenced by the temperature and moisture of the air in the store with the air ¯ows gov- erning the uniformity of the conditions. Improvements in store management systems are being sought and will require a better knowledge of the temporal and spatial variations of conditions within stores.

Numerous attempts have been made to predict heat and mass transfer processes in stored bulks of food- stu€s. These include one-dimensional models of heat and mass transfer (Bakker-Arkema, Bickert & Morey, 1967; Spencer, 1969; OÕCallaghan, Menzies & Bailey, 1971) and two-dimensional models of air ¯ows alone using ®nite di€erence methods (Brooker, 1961) and ®nite element methods (Marchant, 1976), and heat and moisture transfer in sealed bins (Casada & Young, 1994a,b). Other studies have considered three-dimen- sional problems. Sinicio, Jayas, Muir and Sanderson (1992) predicted three-dimensional air ¯ows using a ®- nite element method and Smith, Jayas, Muir, Alagu- sundaram and Kalbande (1992a,b) used a ®nite element method to predict the streamlines of air through a three- dimensional bulk and then used a one-dimensional ®nite di€erence model to predict temperature and moisture changes of the air and solids along each streamline. This method leads to some problems in interpolating between the streamlines and also implies movement of the solids as the streamlines are moved between successive itera- tions during the drying or storage period. Singh, Leon- ardi and Thorpe (1993) and Khankari, Patankar and Morey (1995) have considered air movement and heat and mass transfer in three-dimensional porous media within sealed containers.

The studies above have been carried out on grain bulks containing small particles within which large temperature and moisture gradients are not expected.

Larger items of produce, such as potatoes, can show signi®cant temperature gradients within individual tu- bers. Lerew and Bakker-Arkema (1978) presented a one- dimensional model for the heat and moisture transfer in a column of spherical food materials, e.g. potatoes and fruits, where internal temperature gradients exist within the materials.

In this paper, a transient three-dimensional heat and moisture transfer model, with air ¯ow in an uncon®ned

space of spherical particulate foodstu€s, is presented.

The product conditions (temperature and moisture content) and its properties (conductivity and moisture di€usivity) vary spatially and are used to predict the local heat and mass transfer processes. The global mi- gration of the heat and moisture throughout the storage volume are simulated by solving the Navier±Stokes equations of mass, momentum and energy. This model was tested against experimental results obtained in a bed of potatoes during cooling.

2. The CFD model

To carry out a CFD simulation, the entire store of foodstu€s is subdivided into imaginary control volumes called cells. The cells in the space around the bulk of foodstu€s contain only air whereas the cells inside the bulk contain both air and solid. This mixture of air and solids is treated as a porous medium to predict the air

¯ows. Within each cell, the transfers of heat and mois- ture within a single solid particle and evaporation on its surface are predicted. That particle is considered as representative of all the particles within that particular cell. The following sections describe the equations for the transfers of air, heat and moisture between the cells and within the particles.

2.1. Air ¯ow (air/water vapour mixture)

For homogeneous porous media with constant po- rosity, the continuity and momentum equations de- scribing the air ¯ow are:

o…cq

†

ot ‡ r…q

cu

† ˆ m; …1†

o…cq

u

†

ot ‡ r…cq

u

u

ÿ ci

† ˆ ÿc oP

oX

ÿ R

; …2†

where m is the mass transfer rate per unit volume of a porous medium and is calculated later and R

is the Darcy coecient for multi-dimensional ¯ow through a porous bulk (Ergun, 1952).

2.2. Heat transfer between control volumes

Heat is transferred through a porous medium by the movement of the air and by conduction through the air.

Some heat is also transferred by conduction from the solid particles in one cell to the solid in neighbouring volumes. In mathematical terms

o…qH†

ot ‡ r…cq

u

H

ÿ ck

rT

† ˆ r…k

rT

†; …3†

where subscript bp indicates the bulk properties of the solid particles, T

is the mass-averaged temperature of the solid particles inside a cell, H is the total enthalpy of

p individual solid particle, e.g. a potato ps surface of a solid particle

w water

a porous cell and is the sum of the enthalpy of the solid particles and the air

qH ˆ cq

H

‡ …1 ÿ c†q

H

: …4†

The right-hand-side of Eq. (3) represents the heat con- ducted through the solids.

Substituting Eq. (4) into Eq. (3) and rearranging gives the energy equation for the air ¯ow as

o…cq

H

†

ot ‡ r…cq

u

H

ÿ ck

rT

†

ˆ r…k

rT

† ÿ o‰…1 ÿ c†q

H

Š

ot : …5†

2.3. Moisture transfer between control volumes

The moisture in the air is transferred between neighbouring cells by the movement of the air and by di€usion, as given by

o…cq

Y †

ot ‡ r…cq

u

Y ÿ cq

DrY † ˆ m; …6†

where Y is the mass fraction of vapour and D is the di€usivity of vapour in air.

2.4. Moisture and heat transfer within a single particle A single particle was considered as a sphere consisting of imaginary concentric shells with no variation of the air conditions around the individual particle over a speci®c time interval. Gradients of temperature and moisture content were formed across the surface of the particle and within the particle due to heat and mass transfer at the surface.

The transfer of moisture within the particle is by di€usion such that

ov ot ˆ 1

r

o

or D

r

ov or

 

 

; …7†

where v is the water content and D

is the di€usivity of water inside solid particles.

When there is no condensation, the rate of mass transferred between the solids and air ¯ow per unit volume of porous medium is related to the change of water content within the particle and, assuming no shrinkage of the particle, is given by

m ˆ q

q

V

o ot

Z

Vp

q

v dV ; …8†

where V

is the volume of a particle.

Heat transfer within the particle is due to heat con- duction and the di€usion of moisture

o…q

H

†

ot ˆ qq

‡ 1 r

o

or r

c

q

D

T

ov or

 

 

‡ 1

r

o

or r

k

oT

or

 

 

; …9†

where T

is the temperature inside a particle and qq

includes the heat of respiration and the heat conducted between cells (the term on the left hand side of Eq.

(3)).

2.5. Mass and heat transfer at the surface of a particle The boundary conditions at the surface of the particle show a balance between the moisture and heat reaching and leaving the surface of the particles.

Both forced and natural convections could occur in- side one storage environment. Sherwood number Sh and Nusselt number Nu of the heat and mass transfer on a particle surface during forced convection are calculated using correlations given by Whitaker (1972):

Sh ˆ …0:5Re

‡ 0:2Re

†Sc

…10†

Nu ˆ …0:5Re

‡ 0:2Re

†Pr

…11†

for 10<Re<10

, where the Reynolds number Re is calculated using the characteristic length of the packed bed L

. The correlations used for natural convection were given by Bejan (1995) as

Sh ˆ 0:362Ra

; …12†

Nu ˆ 0:362Ra

: …13†

The higher Nusselt number between Eqs. (11) and (13) and the associated Sherwood number are used for cal- culating the local heat and mass transfer.

The moisture ¯ux F

, evaporated and di€used across the boundary layer on a particle surface into the air

¯ow, is a boundary condition of Eq. (7) and given by F

ˆ h

A…q

ÿ q

† ˆ Sh D A…Y

ÿ Y †q

d

; …14†

where A is the surface area of a particle, h

is mass transfer coecient and Y

is the mass fraction of moisture at the surface of the particle. The value of Y

depends on the relative rates of di€usion of moisture from within the particle to the surface and the rate of evaporation of moisture at the surface. To carry out the calculation of Y

requires a relationship between the water and vapour concentrations at the surface, the sorption isotherm.

The heat ¯ux across the boundary layer on a particle surface F

, which is a boundary condition of Eq. (9), is the sum of the heat transferred due to the temperature di€erence between air and the particle surface and the heat associated with the mass transfer

F

ˆ h

A…T

ÿ T

† ÿ F

…h ‡ c

T

ÿ c

T

†; …15†

where h

is heat transfer coecient and T

the surface

temperature of the particle.

2.6. Solution of momentum, continuity, mass and heat transfer equations

Heat and mass transfer inside foodstu€s during storage are transient processes. During the simulation, the air ¯ow and energy and moisture Eqs. (1)±(6) are solved using the computational ¯uid dynamics package CFDS-CFX4 (AEA Technology, Harwell, Oxford, UK) and then the local heat and mass transfer rates are up- dated from the results of Eqs. (7) and (9) at the end of each time step. These transfer rates are then used in Eqs.

(1)±(6) in the next time step. The number and shape of control volumes used to discretise the stores of food- stu€s and individual particles is dependent on the layout of the store and dimensions of the foodstu€s.

3. A test case: cooling a bed of potatoes 3.1. Experiment details

Experimental data are available on the forced air cooling of an insulated column of new potatoes (Fig. 1) from an initial temperature of 15.5°C using air at 6.7°C, 60% relative humidity and 9.2 m

h

(Misener & Shove, 1976a). The column has a diameter of 0.7 m and a height of 3 m and the height of the bed of packed potatoes is 2.4 m. The bed of potatoes has a bulk density of 390 kgm

and a porosity of 0.61. Misener and Shove (1976b) measured an average length of 95 mm and di- ameter of 51 mm for these tubers, which have surface areas equivalent to the spherical tubers with a diameter of 65 mm. The average velocity through the interstitial spaces between the potatoes is 0.0109 ms

and laminar

¯ow is assumed. The initial water content of the pota- toes, v

, was not reported and 80%, a typical value for new potatoes, is used in the model. Air temperature was

measured up through the bed of potatoes after cooling for 24, 72 and 92 h and the weight of the bed was measured before and after the test.

In the model, axial symmetry was assumed and the simulations were carried out for a radial section of the bed consisting of 256 (32´8) control volumes. Any di€erence in voidage of the bed near to the wall com- pared to that in the bulk was not considered in the model. Uniform velocity, temperature and moisture pro®les were speci®ed at the inlet.

3.2. Properties of the potato

The water di€usivity inside potatoes, D

, is a function of temperature and water content. The di€usivity used here is a combination of the correlation produced by Ruan, Schmidt, Schmidt and Litch®eld (1991) from NMR measurements while drying potato samples at constant temperature (40°C) and the temperature de- pendent correlation by Simal, Rossello, Berna and Mulet (1994)

D

ˆ 5:128  10

e

exp ÿ 3151:5 1 T





ÿ 1

313:15



: The water di€usivity of the skin of potatoes has not been measured so the sensitivity of the predictions of tem- perature and moisture was considered using various values of di€usivity at the surface. These values were given by D

ˆ D

/400 or D

ˆ D

/40 (Burton, 1972, estimated that peeling results in a 300±500 fold increase in the rate of evaporation from the surface of potato).

There have been many measurements of the heat of respiration of potatoes. Data depend on the variety, maturity, and storage conditions of the potatoes. The following correlation was used in the model (Misener &

Shove, 1976a) q

ˆ aT

‡ b;

where a ˆ 6.99 J kg

h

K

(USDA, 1963) and b ˆ 62.62 J kg

h

(to provide agreement with BurtonÕs (1966) data measured on matured new potatoes).

The speci®c heat capacity (Yamada, 1970) and ther- mal conductivity (Burton, 1972) and the thermal con- ductivity of potato bulks (Brugger, 1979) were calculated from

c

ˆ 904 ‡ 3266v for v > 0:5; k

ˆ 0:58;

k

ˆ k

…1 ÿ c

†:

Sorption isotherm data for potato were taken from Mujumdar (1987) and extrapolated to provide a water activity of unity at a surface water content of 80%. The resulting curve for water activity w is given by

w ˆ 1 for v > 0:8

w ˆ 1 ÿ 0:28…0:8 ÿ v† for 0:3 6 v 6 0:8

Fig. 1. Schematic diagram of the experimental arrangement used by Misener & Shove (1976a,b) to study the cooling of potatoes.

There is uncertainty about the relevance of these equa- tions to applications of storage. MujumdarÕs data refers to mashed potato whereas potatoes in storage are whole and the sorption isotherm should strictly relate to the skin of the potato for which no data is available. To examine the sensitivity of the predictions of temperature and moisture pro®les to variations in sorption data, predictions were also carried out using the following equations:

w ˆ 0:95 for v > 0:8;

w ˆ 0:95 ÿ 0:28…0:8 ÿ v† for 0:3 6 v 6 0:8:

The moisture di€usivity, thermal conductivity, density and viscosity of air and the latent heat of water vapour were taken from Bejan (1993).

4. Results and discussions

4.1. Simulation of the cooling front

Before the cooling test, the potatoes were kept at 15.5°C for 10 days so that the potatoes and surrounding air would be in thermal equilibrium. The simulation was begun with the air and potatoes at 15.5°C and 100%

relative humidity throughout the bed. Cooler, less hu- mid, air (6.7°C and 60% r.h.) was then forced into the bottom of the bed. When the cool air contacted the potatoes at the bottom of the bed, the rate of evapora- tion was high because of the large temperature and moisture di€erences between the air and the potato surfaces. Consequently, a cooling front with large gra- dients of temperature and moisture was predicted to develop (Figs. 2 and 3).

The cooling front was gradually pushed through the bed at a rate determined by the air ¯ow rate and the heat and mass transfer processes. After 6 h, the temperature

in the top of the column was greater than the initial temperature (15.5°C) due to the heat from respiration of the potatoes. After continuous cooling for 24 h, the air temperature at the top of the bed had fallen only slightly, to 15.4°C, because of the small di€erence be- tween the vapour concentration in the air and that on the potato surfaces and because of the e€ect of the heat of respiration of the potatoes. The lowest air tempera- ture, 6.4°C, was found near to the bottom of the column where moisture was evaporated from the potatoes and the potato surface temperature was approaching the local air wet bulb temperature. A cooling zone, where the air temperature dropped below the inlet air tem- perature due to evaporative cooling, had formed in the bottom section of the bed. As the air moved through the bed, the humidity of the air was gradually increased from a relative humidity of 60% at the bottom of the bed to just below 100% at the top (Fig. 3). Throughout the 92 h cooling period, the size of the cooling zone was increasing. The lowest temperature predicted inside the cooling zone was 5.3°C which occurred after 92 h.

Radial gradients of temperature and moisture con- centration also existed because of the slow down of the air ¯ow near to the column wall. However, these radial gradients were very small because no heat was lost through the insulated wall of the column.

4.2. Temperature pro®les and weight loss

Fig. 4(a) shows the temperature pro®les measured by Misener and Shove (1976a) along the centre line of the column and the corresponding predictions obtained with D

ˆ D

/40 and w ˆ 1 at initial water content v

ˆ 0.8. After continuous cooling for 24 h, the predicted temperature gradient agrees well with the measurements along the whole column of potatoes. The air tempera- ture measured at the bottom of the bed had dropped to

Fig. 3. Predicted moisture concentration in the air on the centre line along a bed of potatoes after cooling for various times. The potatoes, initially at 15.5°C, were cooled using air at 6.7°C and 60% relative humidity. The predictions are based on Deffˆ Dp/40 and w ˆ 1 at v0ˆ 0.8.

Fig. 2. Predicted air temperature on the centre line along a bed of potatoes after cooling for various times. The potatoes, initially at 15.5°C, were cooled using air at 6.7°C and 60% relative humidity. The predictions are based on Deffˆ Dp/40 and w ˆ 1 at v0ˆ 0.8.

6°C in the cooling zone and the lowest predicted tem- perature was 6.4°C. The di€erence between measured and predicted temperatures suggests that the evapora- tion rate at the bottom of the bed was slightly under- predicted by the model.

After 72 h, the temperatures were predicted reason- ably along much of the column, but di€erences, up to 1.4°C, were found between measured and predicted data in the bottom third of the bed. The air temperature measured 0.2 m from the bottom of the bed was close to the inlet air temperature and was higher than both the measured and predicted temperature at the same posi- tion after 24 h. This increase of temperature suggests that evaporation had decreased probably due to earlier drying of the potato surfaces. The model did not fully predict this drying and consequently it continues to predict high evaporation rates after 72 h. Changes to the properties of the potato surfaces, such as reducing the

skin water di€usivity to D

/400 or the water activity to 0.95, provides some improvement in the predicted tem- perature pro®les but those changes could not fully ac- count for the higher temperatures measured in the bottom of the column, as shown in Fig. 4(b). In the experimental study, turbulence and possible ¯ow ¯uc- tuations in the inlet cooling air could have been created by a centrifugal fan located upstream of the column of potatoes and this would have led to the high drying rate found at the early stages at the bottom of the bed and consequently lower drying rates in the later stages of the test. The model did not consider these e€ects since in- sucient detail was available to provide a reliable sim- ulation of the experimental arrangement.

The average rates of weight loss (weight loss per hour per square metre of the cross section of the potato bed) over each 24 h period of cooling were predicted (Fig. 5) and show the expected trend with the loss rate de- creasing throughout the test period. The rate of loss over the last 24 h was almost half of that over the ®rst 24 h of the test. This reduction is mainly caused by the lower temperature di€erence between the air and the potato surfaces in the later stages. The predicted total weight loss of the potato bed is shown in Fig. 5 and the value after 92 h was 16.3 kg m

which is within 4.5% of the measured value of 17.1 kg m

.

5. Conclusions

We have demonstrated the capability of the model to predict the temperature and moisture changes of pota- toes during cooling. Some di€erences between measured and predicted temperatures were found near to the inlet of the air into the bed of potatoes and this may have been caused by over drying of the potatoes caused by turbu- lence and possible air speed ¯uctuations in the incoming air. Nonetheless, the predicted temperatures agree with

Fig. 5. Predicted rates of weight loss (averaged over 24 h), variations of the total weight loss of the potato bed with time and the weight loss measured after cooling the column of potatoes for 92 h. Predictions based on Deffˆ Dp/40 and w ˆ 1 at v0ˆ 0.8.

Fig. 4. Comparison of measured and predicted air temperature along a bed of potatoes during cooling: (a) Measurements after cooling for 24, 72 and 92 h and predictions made with Deffˆ Dp/40 and w ˆ 1 at v0ˆ 0.8; (b) Measurements after cooling for 92 h and predictions made with Deffvalues of Dp/40 and Dp/400 and w ˆ 0.95 or 1 at v0ˆ 0.8.

measurements to within 1.4°C and show similar spatial variations. Weight loss, a major factor in the economic success of crop storage, is predicted reliably.

Knowledge of the temporal and spatial variations of conditions within stores of foodstu€s is vital to maintain quality and reduce the risk of disease. The model pre- dicts temperature and moisture changes in agreement with measurements at a level that can realistically be expected of measuring instruments in commercial practice. It is in commercial applications, with more complicated geometries and operating strategies, and where measured data is dicult to obtain, that the model will ®nd wide use. Condensation and the appli- cations of the model validated here in commercial crop stores are being considered in the latest development of the model.

Acknowledgements

The authors wish to thank the UK Ministry of Ag- riculture, Fisheries and Food for support for this work and the Biotechnology and Biological Sciences Research Council for support for underlying developments under the ROPA scheme.

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